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About the book

This handbook is a continuation of the Handbook of Elementary Mathematics by the same author and includes material usually studied in mathematics courses of higher educational institutions.
The designation of this handbook is two fold.

Firstly, it is a reference work in which the reader can find definitions (what is a vector product?) and factual information, such as how to find the surface of a solid of revolution or how to expand a function in a trigonometric series, and so on. Definitions, theorems, rules and formulas (accompanied by examples and practical hints) are readily found by reference to the comprehensive index or table of contents.

Secondly, the handbook is intended for systematic reading. It does not take the place of a textbook and so full proofs are only given in exceptional cases. However, it can well serve as material for a first acquaintance with the subject. For this purpose, detailed explanations are given of basic concepts, such as that of a scalar product (Sec. 104), limit (Secs. 203~206), the differential (Secs. 228-235), or infinite series (Secs. 270, 366-370). All rules are abundantly illustrated with examples, which form an integral part of the hand­book (see Secs. 50-62, 134, 149, 264-266, 369, 422, 498, and others). Explanations indicate how to proceed when a rule ceases to be valid; they also point out errors to be avoided (see Secs. 290, 339, 340, 379, and others).

The theorems and rules are also accompanied by a wide range of explanatory material. In some cases, emphasis is placed on bringing out the content of a theorem to facilitate a grasp of the proof. At other times, special examples are illustrated and the reasoning is such as to provide a complete proof of the theorem if applied to the general case (see Secs. 148, 149, 369, 374). Occasionally, the explanation simply refers the reader to the sections on which the proof is based. Material given in small print may be omitted in a first read­ing however, this does not mean it is not important.

Considerable attention has been paid to the historical background of mathematical entities, their origin and development. This very often helps the user to place the subject matter in its proper perspective. Of particular interest in this respect are Secs. 270, 366 together with Secs. 271, 383, 399, and 400, which, it is hoped, will give the reader a clearer understanding of Taylor’s series than is usually obtainable in a formal exposition. Also, biographical information from the lives of mathematicians has been included where deemed advisable.

Table of Contents

PLANE ANALYTIC GEOMETRY

  1. The Subject of Analytic Geometry 19
  2. Coordinates 20
  3. Rectangular Coordinate System 20
  4. Rectangular Coordinates 21
  5. Quadrants 21
  6. Oblique Coordinate System 22
  7. The Equation of a Line 23
  8. The Mutual Positions of a Line and a Point 24
  9. The Mutual Positions of Two Lines 25
  10. The Distance Between Two Points 25
  11. Dividing a Line-Segment in a Given Ratio 26
    1la. Midpoint of a Line-Segment
  12. Second-Order Determinant
  13. The Area of a Triangle
  14. The Straight Line. An Equation Solved for the Ordinate (Slope-
    Intercept Form) 28
  15. A Straight Line Parallel to an Axis 30
  16. The General Equation of the Straight Line 31
  17. Constructing a Straight Line on the Basis of ItsEquation 32
  18. The Parallelism Condition of Straight Lines 32
  19. The Intersection of Straight Lines 34
  20. The Perpendicularity Condition of Two StraightLines 35
  21. The Angle Between Two Straight Lines 36
  22. The Condition for Three Points Lying on OneStraight Line 38
  23. The Equation of a Straight Line Through Two Points (Two-Point Form) 39
  24. A Pencil of Straight Lines 40
  25. The Equation of a Straight Line Through a Given Point and Parallel to a Given Straight Line (Point-Slope Form) 42
  26. The Equation of a Straight Line Through a Given Point and Perpendicular to a Given Straight Line 43
  27. The Mutual Positions of a Straight Line and aPair of Points 44
  28. The Distance from a Point to a Straight Line 44
  29. The Polar Parameters (Coordinates) of a Straight Line 45
  30. The Normal Equation of a Straight Line 47
  31. Reducing the Equation of a Straight Line to the Normal Form 48
  32. Intercepts 49
  33. Intercept Form of the Equation of a Straight Line 50
  34. Transformation of Coordinates (Statement of theProblem) 51
  35. Translation of the Origin 52
  36. Rotation of the Axes 53
  37. Algebraic Curves and Their Order 54
  38. The Circle 56
  39. Finding the Centre and Radius of a Circle 57
  40. The Ellipse as a Compressed Circle 58
  41. An Alternative Definition of the Ellipse 60
  42. Construction of an Ellipse from the Axes 62
  43. The Hyperbola 63
  44. The Shape of the Hyperbola, Its Vertices andAxes 65
  45. Construction of a Hyperbola from Its Axes 67
  46. The Asymptotes of a Hyperbola 67
  47. Conjugate Hyperbolas 68
  48. The Parabola 69
    49 Construction of a Parabola from a Given Parameter p 70
  49. The Parabola as the Graph of the Equation y = ax^{2} + bx + c 70
  50. The Directrices of the Ellipse and of the Hyperbola 73
  51. A General Definition of the Ellipse, Hyperbola and Parabola 75
  52. Conic Sections 77
  53. The Diameters of a Conic Section 78
  54. The Diameters of an Ellipse 79
  55. The Diameters of a Hyperbola 80
  56. The Diameters of a Parabola 82
  57. Second-Order Curves (Quadric Curves) 83
  58. General Second-Degree Equation 85
  59. Simplifying a Second-Degree Equation. General Remarks 86
  60. Preliminary Transformation of a Second-Degree Equation 86
  61. Final Transformation of a Second-Degree Equation 88
  62. Techniques to Facilitate Simplification of a Second-Degree Equation 95
  63. Test for Decomposition of Second-Order Curves 95
    65 Finding Straight Lines that Constitute a Decomposable Second-Order Curve 97
  64. Invariants of a Second-Degree Equation 99
  65. Three Types of Second-Order Curves 102
  66. Central and Noncentral Second-Order Curves (Conics) 104
  67. Finding the Centre of a Central Conic 105
  68. Simplifying the Equation of a Central Conic 107
  69. The Equilateral Hyperbola as the Graph of the Equation y= k/x 109
  70. The Equilateral Hyperbola as the Graph of the Equation
    y = (mx + n)/(px + q) 110
  71. Polar Coordinates 112
  72. Relationship Between Polar and Rectangular Coordinates 114
  73. The Spiral of Archimedes 116
  74. The Polar Equation of a Straight Line 118
  75. The Polar Equation of a Conic Section 119

SOLID ANALYTIC GEOMETRY

  1. Vectors and Scalars. Fundamentals 120
  2. The Vector in Geometry 120
  3. Vector Algebra 121
  4. Collinear Vectors 121
  5. The Null Vector 122
  6. Equality of Vectors 122
  7. Reduction of Vectors to a Common Origin 123
  8. Opposite Vectors 123
  9. Addition of Vectors 123
  10. The Sum of Several Vectors 125
  11. Subtraction of Vectors 126
  12. Multiplication and Division of a Vector by a Number 127
  13. Mutual Relationship of Collinear Vectors (Division of a Vector
    by a Vector) 128
  14. The Projection of a Point on an Axis 129
  15. The Projection of a Vector on an Axis 130
  16. Principal Theorems on Projections of Vectors 132
  17. The Rectangular Coordinate System in Space 133
  18. The Coordinates of a Point 134
  19. The Coordinates of a Vector 135
  20. Expressing a Vector in Terms of Components and in Terms of
    Coordinates 137
  21. Operations Involving Vectors Specified by their Coordinates 137 99. Expressing a Vector in Terms of the Radius Vectors of Its Origin and Terminus 137
  22. The Length of a Vector. The Distance Between Two Points 138
    101 The Angle Between a Coordinate Axis and aVector 139
  23. Criterion of Collinearity (Parallelism) of Vectors 139
  24. Division of a Segment in a Given Ratio 140
  25. Scalar Product of Two Vectors 141
    104a. The Physical Meaning of a Scalar Product 142
  26. Properties of a Scalar Product 142
  27. The Scalar Products of Base Vectors 144
  28. Expressing a Scalar Product in Terms of the Coordinates of the Factors 145
  29. The Perpendicularity Condition of Vectors 146
  30. The Angle Between Vectors 146
  31. Right-Handed and Left-Handed Systems ofThree Vectors 147
  32. The Vector Product of Two Vectors 148
  33. The Properties of a Vector Product 150
  34. The Vector Products of the Base Vectors 152
  35. Expressing a Vector Product in Terms of the Coordinates of
    the Factors 152
  36. Coplanar Vectors 154
  37. Scalar Triple Product 154
    117 Properties of a Scalar Triple Product 155
  38. Third-Order Determinant 156
  39. Expressing a Triple Product in Terms of the Coordinates of the
    Factors 169
  40. Coplanarity Criterion in Coordinate Form 159
  41. Volume of a Parallelepiped 160
  42. Vector Triple Product 161
  43. The Equation of a Plane 161
  44. Special Cases of the Position of a Plane Relative to a Coordi­nate System 162
  45. Condition of Parallelism of Planes 163
  46. Condition of Perpendicularity of Planes 164
  47. Angle Between Two PlaneS 164
  48. A Plane Passing Through a Given Point Parallel to a Given Plane 165
  49. A Plane Passing Through Three Points 165
  50. Intercepts on tne Axes 166
  51. Intercept Form of the Equation of a Plane 166
  52. A Plane Passing Through Two Points Perpendicular to a Given Plane 167
  53. A Plane Passing Through a Given Point Perpendicular to Two Planes 167
  54. The Point of Intersection of Three Planes 168
  55. The Mutual Positions of a Plane and a Pair of Points 169
  56. The Distance from a Point to a Plane 170
  57. The Polar Parameters (Coordinates) of a Plane 170
  58. The Normal Equation of a Plane 172
  59. Reducing the Equation of a Plane to the Normal Form 173
  60. Equations of a Straight Line in Space 174
  61. Condition Under Which Two First-Degree Equations Represent a Straight Line 176
  62. The Intersection of a Straight Line and a Plane 177
  63. The Direction Vector 179
  64. Angles Between a Straight Line and the Coordinate Axes 179 145. Angle Between Two Straight Lines 180 146. Angle Between a Straight Line and a Plane 181
  65. Conditions of Parallelism and Perpendicularity of a Straight Line and a Plane 181
  66. A Pencil of Planes 182
  67. Projections of a Straight Line on the Coordinate Planes 184
  68. Symmetric Form of the Equation of a Straight Line 185
  69. Reducing the Equations of a Straight Line to Symmetric Form 187
  70. Parametric Equations of a Straight Line 188
  71. The Intersection of a Plane with a Straight Line Represented Parametrically 189
  72. The Two-Point Form of the Equations of a Straight Line 190
  73. The Equation of a Plane Passing Through a Given Point Perpendicular to a Given Straight Line 190
  74. The Equations of a Straight Line Passing Through a Given Point Perpendicular to a Given Plane 190
  75. The Equation of a Plane Passing Through a Given Point and a Given Straight Line 191
  76. The Equation of a Plane Passing Through a Given Point Parallel to Two Given Straight Lines 192
  77. The Equation of a Plane Passing Through a Given Straight Line and Parallel to Another Given Straight Line 192
  78. The Equation of a Plane Passing Through a Given Straight Line and Perpendicular to a Given Plane 193
  79. The Equations of a Perpendicular Dropped from a Given Point onto a Given Straight Line 193
  80. The Length of a Perpendicular Dropped from a Given Point onto a Given Straight Line 195
  81. The Condition for Two Straight Lines Intersecting or Lying in a Single Plane 196
  82. The Equations of a Line Perpendicular to Two Given Straight Lines 197
  83. The Shortest Distance Between Two Straight Lines 199
    165a. Right-Handed and Left-Handed Pairs of Straight Lines 201
  84. Transformation of Coordinates 202
  85. The Equation of a Surface 203 168. Cylindrical Surfaces Whose Generatrices Are Parallel to One of the Coordinate Axes 204
  86. The Equations of a Line 205
  87. The Projection of a Line on a Coordinate Plane 206
  88. Algebraic Surfaces and Their Order 209
  89. The Sphere 209
  90. The Ellipsoid 210
  91. Hyperboloid of One Sheet 213
  92. Hyperboloid of Two Sheets 215
  93. Quadric Conical Surface 217
  94. Elliptic Paraboloid 218
  95. Hyperbolic Paraboloid 220
  96. Quadric Surfaces Classified 221
  97. Straight-Line Generatrices of Quadric Surfaces 224
  98. Surfaces of Revolution 225
  99. Determinants of Second and Third Order 226
  100. Determinants of Higher Order 229
  101. Properties of Determinants 231 185. A Practical Technique for Computing Determinants 233
  102. Using Determinants to Investigate and Solve Systems of Equations 236
  103. Two Equations in Two Unknowns 236
  104. Two Equations in Three Unknowns 238
  105. A Homogeneous System of Two Equations in Three Unknowns 240
    190 Three Equations in Three Unknowns 241
    190a. A System of n Equations in n Unknowns 246

FUNDAMENTALS OF MATHEMATICAL ANALYSIS

  1. Introductory Remarks 247
  2. Rational Numbers 248
  3. Real Numbers 248
  4. The Number Line 249
  5. Variable and Constant Quantities 250
  6. Function 250
  7. Ways of Representing Functions 252
  8. The Domain of Definition of a Function 254
  9. Intervals 257
  10. Classification of Functions 258
  11. Basic Elementary Functions 259
  12. Functional Notation 259
  13. The Limit of a Sequence 261
  14. The Limit of a Function 262
  15. The Limit of a Function Defined 264
  16. The Limit of a Constant 265
  17. Infinitesimals 265
  18. Infinities 266
  19. The Relationship Between Infinities and Infinitesimals 267
  20. Bounded Quantities 267
  21. An Extension of the Limit Concept 267
  22. Basic Properties of Infinitesimals 269
  23. Basic Limit Theorems 270
  24. The Number e 271
  25. The Limit of sin x / x as x → 0 273
  26. Equivalent Infinitesimals 273
  27. Comparison of Infinitesimals 274
    217a. The Increment of a Variable Quantity 276
  28. The Continuity of a Function at a Point 277
  29. The Properties of Functions Continuous at a Point 278
    219a. One-Sided (Unilateral) Limits. The Jump of a Function 278
  30. The Continuity of a Function on a Closed Interval 279
  31. The Properties of Functions Continuous on a Closed Interval 280

DIFFERENTIAL CALCULUS

  1. Introductory Remarks 282
  2. Velocity 282
  3. The Derivative Defined 284
  4. Tangent Line 285
  5. The Derivatives of Some Elementary Functions 287
  6. Properties of a Derivative 288
  7. The Differential 289
  8. The Mechanical Interpretation of a Differential 290
  9. The Geometrical Interpretation of a Differential 291
  10. Differentiable Functions 291
  11. The Differentials of Some Elementary Functions 294
  12. Properties of a Differential 294
  13. The Invariance of the Expression f'(x) dx 294
  14. Expressing a Derivative in Terms of Differentials 295
  15. The Function of a Function (Composite Function) 296
  16. The Differential of a Composite Function 296
  17. The Derivative of a Composite Function 297
  18. Differentiation of a Product 298
  19. Differentiation of a Quotient (Fraction) 299
  20. Inverse Function 300
  21. Natural Logarithms 302
  22. Differentiation of a Logarithmic Function 303
  23. Logarithmic Differentiation 304
  24. Differentiating an Exponential Function 306
  25. Differentiating Trigonometrie Functions 307
  26. Differentiating Inverse Trigonometrie Functions 308
    247a. Some Instructive Examples 309
  27. The Differential in Approximate Calculations 311
  28. Using the Differential to Estimate Errors in Formulas 318
  29. Differentiation of Implicit Functions 315
  30. Parametric Representation of a Curve 316
  31. Parametric Representation of a Function 318
  32. The Cycloid 320
  33. The Equation of a Tangent Line to a Plane Curve 321
    254a. Tangent Lines to Quadric Curves 323
  34. The Equation of a Normal 323
  35. Higher-Order Derivatives 324
  36. Mechanical Meaning of the Second Derivative 325
  37. Higher-Order Differentials 326
  38. Expressing Higher Derivatives in Terms of Differentials 329
  39. Higher Derivatives of Functions Represented Parametrically 330 261. Higher Derivatives of Implicit Functions 331
  40. Leibniz Rule 332
  41. Rolle’s Theorem 334
  42. Lagrange’s Mean-Value Theorem 335
  43. Formula of Finite Increments 337
  44. Generalized Mean-Value Theorem (Cauchy) 339
  45. Evaluating the Indeterminate Form 0/0 341
  46. Evaluating the Indeterminate Form ∞/∞ 344
  47. Other indeterminate Expressions 345
  48. Taylor’s Formula (Historical Background) 347
  49. Taylor’s Formula 351
  50. Taylor’s Formula for Computing the Values of a Function 353
  51. Increase and Decrease of a Function 360
  52. Tests for the Increase and Decrease of a Function at a Point 362 274a. Tests for the Increase and Decrease of a Function in an Interval 363
  53. Maxima and Minima 364
  54. Necessary Condition for a Maximum and a Minimum 365
  55. The First Sufficient Condition for a Maximum and a Minimum 366 278. Rule for Finding Maxima and Minima 366
  56. The Second Sufficient Condition for a Maximum and a Minimum 372 280. Finding Greatest and Least Values of a Function 372
  57. The Convexity of Plane Curves. Point of Inflection 379
  58. Direction of Concavity 380
  59. Rule for Finding Points of Inflection 381
  60. Asymptotes 383
  61. Finding Asymptotes Parallel to the CoordinateAxes 383
  62. Finding Asymptotes Not Parallel to the Axis ofOrdinates 386
  63. Construction of Graphs (Examples) 388
  64. Solution of Equations. General Remarks 392
  65. Solution of Equations. Method of Chords 394
  66. Solution of Equations. Method of Tangents 396
  67. Combined Chord and Tangent Method 398

INTEGRAL CALCULUS

  1. Introductory Remarks 401
  2. Antiderivative 403
  3. Indefinite Integral 404
  4. Geometrical Interpretation of Integration 406
  5. Computing the Integration Constant from Initial Data 409
  6. Properties of the Indefinite Integral 410
  7. Table of Integrais 411
  8. Direct integration 413
  9. Integration by Substitution (Change of Variable) 414
  10. Integration by Parts 418
  11. Integration of Some Trigonometric Expressions 421
  12. Trigonometrie Substitutions 426
  13. Rational Functions 426
    304a. Taking out the Integral Part 426
  14. Techniques for Integrating Rational Fractions 427
  15. Integration of Partial Rational Fractions 428
  16. Integration of Rational Functions (General Method) 431
  17. Factoring a Polynomial 438
  18. On the Integrability of Elementary Functions 439
  19. Some Integrais Dependent on Radicals 439
  20. The Integral of a Binomial Differential 441
  21. Integrais of the Form ∫ R (x, √(ax^{2} + bx + c) dx 443
  22. Integrais of the Form ∫ R (sin x, cos x) dx 445
  23. The Definite Integral 446
  24. Properties of the Definite Integral 450
  25. Geometrical Interpretation of the Definite Integral 452
  26. Mechanical Interpretation of the Definite Integral 453
  27. Evaluating a Definite Integral 455
    318a. The Bunyakovsky Inequality 456
  28. The Mean-Value Theorem of Integral Calculus 456
  29. The Definite Integral as a Function of the Upper Limit 458
  30. The Differential of an Integral 460
  31. The Integral of a Differential. The Newton-Leibniz Formula 462 323. Computing a Definite Integral by Means of the Indefinite
    Integral 464
  32. Definite Integration by Parts 465
  33. The Method of Substitution in a Definite Integral 466
  34. On Improper Integrais 471
  35. Integrais with Infinite Limits 472
  36. The Integral of a Function with a Discontinuity 476
  37. Approximate Integration 480
  38. Rectangle Formulas 483
  39. Trapezoid Rule 485
  40. Simpson’s Rule (for Parabolic Trapezoids) 486
  41. Areas of Figures Referred to Rectangular Coordinates 488
  42. Scheme for Employing the Definite Integral 490
  43. Areas of Figures Referred to Polar Coordinates 492
  44. The Volume of a Solid Computed by the Shell Method 494
  45. The Volume of a Solid of Revolution 496
  46. The Arc Length of a Plane Curve 497
  47. Differential of Arc Length 499
  48. The Arc Length and Its Differential inPolarCoordinates 499
  49. The Area of a Surface of Revolution 501

PLANE AND SPACE CURVES (FUNDAMENTALS)

  1. Curvature 503
  2. The Centre, Radius and Circle of Curvature of a Plane Curve 504
  3. Formulas for the Curvature, Radius and Centre of Curvature of a Plane Curve 505
  4. The Evolute of a Plane Curve 508
  5. The Properties of the Evolute of a Plane Curve 510
  6. Involute of a Plane Curve 511
  7. Parametric Representation of a Space Curve 512
  8. Helix 514
  9. The Arc Length of a Space Curve 515
  10. A Tangent to a Space Curve 516
  11. Normal Planes 518
  12. The Vector Function of a Scalar Argument 519
  13. The Limit of a Vector Function 520
  14. The Derivative Vector Function 521
  15. The Differential of a Vector Function 523
  16. The Properties of the Derivative and Differential of a Vector Function 524
  17. Osculating Plane 525
  18. Principal Normal. The Moving Trihedron 527
  19. Mutual Positions of a Curve and a Plane 529
  20. The Base Vectors of the Moving Trihedron 529
  21. The Centre, Axis and Radius of Curvature of a Space Curve 530
  22. Formulas for the Curvature, and the Radius and Centre of Cur­vature of a Space Curve 531
  23. On the Sign of the Curvature 534
  24. Torsion 535

SERIES

  1. Introductory Remarks 637
  2. The Definition of a Series 537
  3. Convergent and Divergent Series 538
  4. A Necessary Condition for Convergence of a Series 540
  5. The Remainder of a Series 542
  6. Elementary Operations on Series 543
  7. Positive Series 545
  8. Comparing Positive Series 545
  9. D’Alembert’s Test for a Positive Series 548
  10. The Integral Test for Convergence 549
  11. Alternating Series. Leibniz’ Test 552
  12. Absolute and Conditional Convergence 553
  13. D’Alembert’s Test for an Arbitrary Series 555
  14. Rearranging the Terms of a Series 555
  15. Grouping the Terms of a Series 556
  16. Multiplication of Series 558
  17. Division of Series 561
  18. Functional Series 562
  19. The Domain of Convergence of a Functional Series 563
  20. On Uniform and Nonuniform Convergence 565
  21. Uniform and Nonuniform Convergence Defined 568
  22. A Geometrical Interpretation of Uniform and Nonuniform Con­vergence 568
  23. A Test for Uniform Convergence. Regular Series 569
  24. Continuity of the Sum of a Series 570
  25. Integration of Series 571
  26. Differentiation of Series 575
  27. Power Series 576
  28. The Interval and Radius of Convergence of a Power Series 577
  29. Finding the Radius of Convergence 578
  30. The Domain of Convergence of a Series Arranged in Powers of x – x_{0} 580
  31. Abel’s Theorem 581
  32. Operations on Power Series 582
  33. Differentiation and Integration of a Power Series 584
  34. Taylor’s Series 586
  35. Expansion of a Function in a Power Series 587
  36. Power-Series Expansions of Elementary Functions 589
  37. The Use of Series in Computing Integrais 594
  38. Hyperbolic Functions 595
  39. Inverse Hyperbolic Functions 598
  40. On the Origin of the Names of the Hyperbolic Functions 600
  41. Complex Numbers 601
  42. A Complex Function of a Real Argument 602
  43. The Derivative of a Complex Function 604
  44. Raising a Positive Number to a Complex Power 605
  45. Euler’s Formula 607
  46. Trigonometrie Series 608
  47. Trigonometrie Series (Historical Background) 608
  48. The Orthogonality of the System of Functions cos nx, sin nx 609 414. Euler-Fourier Formulas 611
  49. Fourier Series 615
  50. The Fourier Series of a Continuous Function 615
  51. The Fourier Series of Even and Odd Functions 618
  52. The Fourier Series of a Discontinuous Function 622

DIFFERENTIATION AND INTEGRATION OF FUNCTIONS OF SEVERAL VARIABLES

  1. A Function of Two Arguments 626
  2. A Function of Three and More Arguments 627
  3. Modes of Representing Functions of Several Arguments 628
  4. The Limit of a Function of Several Arguments 630
  5. On the Order of Smallness of a Function of Several Arguments 632 424. Continuity of a Function of Several Arguments 633
  6. Partial Derivatives 634
  7. A Geometrical Interpretation of Partial Derivatives for the Case of Two Arguments 635
  8. Total and Partial Increments 636
  9. Partial Differential 636
  10. Expressing a Partial Derivative in Terms of a Differential 637
  11. Total Differential 638
  12. Geometrical Interpretation of the Total Differential (for the Case of Two Arguments) 640
  13. Invariance of the differential Expression f’x dx +f’y dy +f’z dz
    of the Total Di­fferential 640
  14. The Technique of Differentiation 641
  15. Differentiable Functions 642
  16. The Tangent Plane and the Normal to a Surface 643
  17. The Equation of the Tangent Plane 644
  18. The Equation of the Normal 646
  19. Differentiation of a Composite Function 646
  20. Changing from Rectangular to Polar Coordinates 647
  21. Formulas for Derivatives of a Composite Function 648
  22. Total Derivative 649
  23. Differentiation of an Implicit Function of Several Variables 650 443. Higher-Order Partial Derivatives 653
  24. Total Differentials of Higher Orders 654
  25. The Technique of Repeated Differentiation 656
  26. Symbolism of Differentials 657
  27. Taylor’s Formula for a Function of Several Arguments 658
  28. The Extremum (Maximum or Minimum) of a Function of Seve­ral Arguments 660
  29. Rule for Finding an Extremum 660
  30. Sufficient Conditions for an Extremum (for the Case of Two Arguments) 662
  31. Double Integral 663
  32. Geometrical Interpretation of a Double Integral 665
  33. Properties of a Double Integral 666
  34. Estimating a Double Integral 666
  35. Computing a Double Integral (Simplest Case) 667
  36. Computing a Double Integral (General Case) 670
  37. Point Function 674
  38. Expressing a Double Integral in Polar Coordinates 675
  39. The Area of a Piece of Surface 677
  40. Triple Integral 681
  41. Computing a Triple Integral (Simplest Case) 681
  42. Computing a Triple Integral (General Case) 682
  43. Cylindrical Coordinates 685
  44. Expressing a Triple Integral in Cylindrical Coordinates 685
  45. Spherical Coordinates 686
  46. Expressing a Triple Integral in Spherical Coordinates 687
  47. Scheme for Applying Double and Triple Integrais 688
  48. Moment of Inertia 689
  49. Expressing Certain Physical and Geometrical Quantities in Terms of Double Integrais 691
  50. Expressing Certain Physical and Geometrical Quantities in Terms of Triple Integrals 693
  51. Line Integrals 695
  52. Mechanical Meaning of a Line Integral 697
  53. Computing a Line Integral 698
  54. Green’s Formula 700
  55. Condition Under Which Line Integral Is Independent of Path 701
  56. An Alternative Form of the Condition Given in Sec. 475 703

DIFFERENTIAL EQUATIONS

  1. Fundamentals 706
  2. First-Order Equation 708
  3. Geometrical Interpretation of a First-Order Equation 708
  4. Isoclines 711
  5. Particular and General Solutions of a First-Order Equation 712
  6. Equations with Variables Separated 713
  7. Separation of Variables. General Solution 714
  8. Total Differential Equation 716 484a. Integrating Factor 717
  9. Homogeneous Equation 718
  10. First-Order Linear Equation 720
  11. Clairaut’s Equation 722
  12. Envelope 724
  13. On the Integrability of Differential Equations 726
  14. Approximate Integration of First-Order Equations by Euler’s Method 726
  15. Integration of Differential Equations by Means of Series 728
  16. Forming Differential Equations 730
  17. Second-Order Equations 734
  18. Equations of the nth Order 736
  19. Reducing the Order of an Equation 736
  20. Second-Order Linear Differential Equations 738
  21. Second-Order Linear Equations with Constant Coefficients 742
  22. Second-Order Homogeneous Linear Equations with Constant Coefficients 742
    498a. Connection Between Cases 1 and 3 in Sec. 498 744
  23. Second-Order Nonhomogeneous Linear Equations with Constant Coefficients 744
  24. Linear Equations of Any Order 750
  25. Method of Variation of Constants (Parameters) 752
  26. Systems of Differential Equations. Linear Systems 754

SOME REMARKABLE CURVES

  1. Strophoid 756
  2. Cissoid of Diodes 758
  3. Leaf of Descartes 760
  4. Versiera 763
  5. Conchoid of Nicomedes 766
  6. Limaçon. Cardioid 770
  7. Cassinian Curves 774
  8. Lemniscate of Bernoulli 779
  9. Spiral of Archimedes 782
  10. Involute of a Circle 785
  11. Logarithmic Spiral 789
  12. Cycloids 795
  13. Epicycloids and Hypocycloids 810
  14. Tractrix 826
  15. Catenary 833

TABLES

I. Natural Logarithms 839
II. Table for Changing from Natural Logarithms to Common Lo­garithms 843
III. Table for Changing from Common Logarithms to Natural Loga­rithms
IV. The Exponential Function e^{x} 844
V. Table of Indefinite Integrals 846
Index 854

Originally posted at Mathematical Handbook – Higher Mathematics – Vygodsky